International Research Journal of Engineering and Technology (IRJET)
e-ISSN: 2395-0056
Volume: 03 Issue: 01 | Jan-2016
p-ISSN: 2395-0072
www.irjet.net
In insight into QAC2(1) : Dynkin diagrams and properties of roots A.Uma Maheswari Associate Professor, Department of Mathematics, Quaid-E-Millath Government College for Women, Chennai , Tamilnadu, India ---------------------------------------------------------------------***---------------------------------------------------------------------
Abstract - In this paper the indefinite quasi affine Kac
studied by Feingold & Frenkel [4], Benkart , Kang & Misra ([1],); Kang studied the structure of HA1(1), HA2(1) and HAn(1) ([6]-[8]);Other sub classes of the indefinite type, namely Extended hyperbolic were introduced by Sthanumoorthy & Uma Maheswari ([10],[11] ) and determined the structure and computed root multiplicities up to level 5 for EHA1(1) , EHA2(2)( [12]-[14]) ;
Moody algebras QAC2 are considered; basic properties of real and imaginary roots are studied for three specific classes of quasi affine Kac Moody algebras belonging to the family QAC2(1); the short and long real roots are identified; the minimal imaginary roots and isotropic roots up to height 3 have been computed for these classes of Kac Moody algebras; These algebras also satisfy the purely imaginary property i.e. all the imaginary roots are purely imaginary ; All the 909 non isomorphic connected Dynkin diagrams associated with QAC2(1) , are given in the classification theorem. It is also proved that there are 648 Dynkin diagrams of extended hyperbolic type, 8 hyperbolic type and 253 diagrams of indefinite, non extended hyperbolic type in the classification of Dynkin diagrams associated with QAC2(1). (1)
Quasi hyperbolic and Quasi affine types were introduced by Uma Maheswari ([15],[18]). Uma Maheswari and et.al studied Quasi hyperbolic algebras QHG2,QHA2(1), QHA4(2) , QHA7(2), QHA5(2) ([20]-[24]), quasi hyperbolic Dynkin diagrams of rank 3 in [16] and quasi affine Kac Moody algebras QAD3(2) , QAG2(1) ( [18]-[19]). In [17], the family QAC2(1) was defined, the structure of a specific class of QAC2(1) was determined using homological techniques and the general form of the Dynkin diagrams associated with this family QAC2(1) was given; Here in this paper, the complete classification of non isomorphic connected Dynkin diagrams ( 909 in number) associated with QAC2(1) is given; Also, we consider three particular classes belonging to QAC2(1) and study the basic properties of real and imaginary roots for the associated Quasi affine Kac Moody algebras QAC2(1).
Key Words: Kac Moody algebras, Dynkin diagrams, roots, real , imaginary,, quasi hyperbolic, quasi affine. AMS Classification :17B67 1. INTRODUCTION
2. PRELIMINARIES
The subject Kac Moody algebras was developed by Kac and Moody in 1968, simultaneously and independently ([5],[9]). Among the three main types of Kac Moody algebras, namely finite, affine and indefinite, the study of indefinite type of Kac Moody algebras is still wide open, whose structure is not completely understood; The Kac Moody algebras were completely classified through Dynkin digrams for the finite, affine and, hyperbolic types ([5], [25]). The root systems of the finite and affine types have been characterized ([5],[9],[25]); As far as the imaginary roots of the indefinite type Kac Moody algebras are concerned, strictly imaginary roots were studied by Casperson [3] and special imaginary roots by Bennett [2]; Purely imaginary roots were introduced by Sthanumoorthy and Uma Maheswari ([10] ,[11]) ; Among the indefinite Kac Moody algebras, hyperbolic Kac Moody algebras , which are natural extensions of affine type were © 2016, IRJET |
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DEFINITION 1: A realization of a matrix A ( aij )i , j 1 is a n
( H , , ) where l is the rank of A, H is a 2n - l dimensional complex vector space, {1 , 2 ,..., n } triple and
{1 , 2 ,..., n } are linearly independent
subsets of H* and H respectively, satisfying
j ( i ) a ij
for i, j = 1,….,n. is called the root basis. Elements of are called simple roots. The root lattice generated by is Q
n
z i 1
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